In this paper, we will consider a general framework for these and other stepsizes by introducing a harmonic Rayleigh quotient including a target \(\tau \) .
We recall the harmonic Rayleigh–Ritz extraction for matrix eigenvalue problems in Section .
The general form of this extraction features a target \(\tau \in \mathbb {R} \, \cup \, \lbrace  \pm \infty \rbrace \) .
This target is analyzed and exploited in Section , to develop a new general framework for all possible stepsizes. We will see that the BB stepsizes correspond to \(\tau = 0\)  or \(\tau = \pm \infty \) .
This may not only add towards a new understanding and interpretation of known strategies, but also suggests new competitive schemes.
Section  closer studies the Adaptive Barzilai–Borwein method (ABB) [1]}, and provides a new theoretical justification for it. We also showcase the potential of the framework by introducing new families generalizing the ABB method.
As is common (see, e.g., [2]}) we first consider the convex quadratic problem.
Convergence results for this case, extending those of [3]}, are presented in Section .
The extension of the harmonic steplength to general nonlinear problems is treated in Section .
Finally, we carry out numerical experiments and summarize some conclusions in Sections  and .
